Many search engine services, such as Google and Overture, provide for searching for information that is accessible via the Internet. These search engine services allow users to search for display pages, such as web pages, that may be of interest to users. After a user submits a search request (i.e., a query) that includes search terms, the search engine service identifies web pages that may be related to those search terms. To quickly identify related web pages, the search engine services may maintain a mapping of keywords to web pages. This mapping may be generated by “crawling” the web (i.e., the World Wide Web) to identify the keywords of each web page. To crawl the web, a search engine service may use a list of root web pages to identify all web pages that are accessible through those root web pages. The keywords of any particular web page can be identified using various well-known information retrieval techniques, such as identifying the words of a headline, the words supplied in the metadata of the web page, the words that are highlighted, and so on. The search engine service identifies web pages that may be related to the search request based on how well the keywords of a web page match the words of the query. The search engine service then displays to the user links to the identified web pages in an order that is based on a ranking that may be determined by their relevance to the query, popularity, importance, and/or some other measure.
Three well-known techniques for ranking of web pages are PageRank, HITS (“Hyperlinked-Induced Topic Search”), and DirectHIT. PageRank is based on the principle that web pages will have links to (i.e., “outgoing links”) important web pages. Thus, the importance of a web page is based on the number and importance of other web pages that link to that web page (i.e., “incoming links”). In a simple form, the links between web pages can be represented by adjacency matrix A, where Aij represents the number of outgoing links from web page i to web page j. The importance score wj for web page j can be represented by the following equation:wj=ΣiAijwi 
This equation can be solved by iterative calculations based on the following equation:ATw=w where w is the vector of importance scores for the web pages and is the principal eigenvector of AT.
The HITS technique is additionally based on the principle that a web page that has many links to other important web pages may itself be important. Thus, HITS divides “importance” of web pages into two related attributes: “hub” and “authority.” “Hub” is measured by the “authority” score of the web pages that a web page links to, and “authority” is measured by the “hub” score of the web pages that link to the web page. In contrast to PageRank, which calculates the importance of web pages independently from the query, HITS calculates importance based on the web pages of the result and web pages that are related to the web pages of the result by following incoming and outgoing links. HITS submits a query to a search engine service and uses the web pages of the result as the initial set of web pages. HITS adds to the set those web pages that are the destinations of incoming links and those web pages that are the sources of outgoing links of the web pages of the result. HITS then calculates the authority and hub score of each web page using an iterative algorithm. The authority and hub scores can be represented by the following equations:
      a    ⁡          (      p      )        =                    ∑                  q          ->          p                    ⁢                        h          ⁡                      (            q            )                          ⁢                                  ⁢        and        ⁢                                  ⁢                  h          ⁡                      (            p            )                                =                  ∑                  p          ->          q                    ⁢              a        ⁡                  (          q          )                    where a(p) represents the authority score for web page p and h (p) represents the hub score for web page p. HITS uses an adjacency matrix A to represent the links. The adjacency matrix is represented by the following equation:
      b    ij    =      {                            1                                                    if              ⁢                                                          ⁢              page              ⁢                                                          ⁢              i              ⁢                                                          ⁢              has              ⁢                                                          ⁢              a              ⁢                                                          ⁢              link              ⁢                                                          ⁢              to              ⁢                                                          ⁢              page              ⁢                                                          ⁢              j                        ,                                                0                          otherwise                    
The vectors a and h correspond to the authority and hub scores, respectively, of all web pages in the set and can be represented by the following equations:a=ATh and h=Aa Thus, a and h are eigenvectors of matrices ATA and AAT. HITS may also be modified to factor in the popularity of a web page as measured by the number of visits. Based on an analysis of click-through data, bij of the adjacency matrix can be increased whenever a user travels from web page i to web page j.
DirectHIT ranks web pages based on past user history with results of similar queries. For example, if users who submit similar queries typically first selected the third web page of the result, then this user history would be an indication that the third web page should be ranked higher. As another example, if users who submit similar queries typically spend the most time viewing the fourth web page of the result, then this user history would be an indication that the fourth web page should be ranked higher. DirectHIT derives the user histories from analysis of click-through data.
Some ranking techniques use machine learning algorithms to learn a ranking function from training data that includes queries, feature vectors representing pages, and for each query, a ranking for each page. A ranking function serves as a mapping from features of a page to its rank for a given query. The learning of a ranking function has been considered by some as a regression problem for learning the mapping of a feature vector to a member of an ordered set of numerical ranks. Some regression based techniques attempt to provide an absolute relevance score that can be used to rank pages. A ranking function, however, need not provide an absolute relevance score but rather need only provide a relative ranking of the pages. Thus, these regression-based techniques solve a problem that is more difficult than needed.
Machine learning algorithms for a ranking function use queries, feature vectors, and user-labeled relevance scores as training data. To generate the training data, queries may be submitted to a search engine which generates the pages of the search result. The algorithms then generate the feature vectors for the pages and input from a user the relevance scores for each page. A difficulty with such an approach is that a search engine may return hundreds of pages as its search result. It can be quite costly to have a user label all the pages of a search result. Moreover, it can be difficult for a user to accurately assess the relevance of such a large number of pages. Although a user could label only a small portion of the pages, the learning based on such a small portion may not provide an accurate ranking function.